I assert that the Monotone Convergence Theorem can only help us conclude
that the sequence of partial sums converges if we can also show that the
sequence of partial sums is bounded above. Is this true?

Why do we need to include the solution to the homogeneous equation as
part of the solution to a non-homogeneous equation? Since the solution to
the homogeneous part is by definition zero, can't we ignore it?

Take the second derivative of t/exp(t) -1 and find that it is an even
function; show that an even function, when expressed as a Taylor
Series, has only even powers of t; draw the conclusion that the odd
Bernoulli numbers have to be 0.

A student wonders if sech(z) is complex differentiable, and where. Picking up on the
student's familiarity with the Cauchy-Riemann equations, Doctor Jordan uses
trigonometric identities to examine the function's real and imaginary parts separately,
and reveals the conditions under which the function is holomorphic.

I know that velocity components are dr/dt and rdo/dt and that components
of acceleration are d^2r/dt^2 - r(do/dt)^2, etc. I would appreciate any
help at all in understanding how these are generated and/or how to use
them.